# How do you write equations in function notation?

Oct 28, 2014

Most students will be introduced to function notation after studying linear functions for a little while.

For example, $y = 2 x + 3$ is my favorite linear function. We like to be able to spot the slope easily, $m = 2$, and the y-intercept as well, $b = 3$. Graphing is also made simple with this information. This equation is also written as $f \left(x\right) = 2 x + 3$, which means, this function depends on $x$, and is defined by doubling $x$ and adding 3 to the result. Too many words!

However, this notation allows someone to say "find f(1)" which means to evaluate the function when $x = 1$.

(it takes fewer words that way!) So, $f \left(1\right) = 2 \left(1\right) + 3 = 5$.

The ordered pair $\left(1 , 5\right)$ is a point on your graph.

Try some more:

$f \left(0\right) = 2 \left(0\right) + 3 = 3$
$f \left(- 2\right) = 2 \left(- 2\right) + 3 = - 4 + 3 = - 1$
etc.

What if someone said "find $x$ when $f \left(x\right) = 9$".
That means, solve $9 = 2 x + 3$. (replace $f \left(x\right)$ with 9)

$9 - 3 = 2 x + 3 - 3$
$6 = 2 x$
$\frac{6}{2} = 2 \frac{x}{2}$
so $3 = x$!

$\left(3 , 9\right)$ would be the ordered pair on the graph.

As an additional note, I tell students that sometimes you may see different letters than $f \left(x\right)$. "F" stands for function, literally. Other letters could be used like g(x), h(x), etc. for typical algebra problems.

In a science setting, you could encounter something like h(t) which might mean "height with respect to time". Or, height is a function of time.

Perhaps you have seen P(T) which could mean Pressure depends on Temperature. How about d(t), or distance with respect to time? (how far you go depends on how long you have been traveling....)

Soon, you will learn the ABCs of functions!